31:
2267:. As the graph shows, a sundial can be up to about 16 minutes fast or slow, compared with a clock. Since the Earth rotates at a mean speed of one degree every four minutes, relative to the Sun, this 16-minute displacement corresponds to a shift eastward or westward of about four degrees in the apparent position of the Sun, compared with its mean position. A westward shift causes the sundial to be ahead of the clock.
1375:
2241:
3108:
1804:) is also discouraged for having an error of up to 0.28°. An additional error of up to 0.5° can occur in all equations around the equinoxes if not using a decimal place when selecting N to adjust for the time after UT midnight for the beginning of that day. So the above equation can have up to 2.0° of error, about four times the Sun's angular width, depending on how it is used.
1509:, in early July. This makes processes like the variation of the solar declination happen faster in January than in July. On the graph, this makes the minima more acute than the maxima. Also, since perihelion and aphelion do not happen on the exact dates as the solstices, the maxima and minima are slightly asymmetrical. The rates of change before and after are not quite equal.
1800:−1). The number 10, in (N+10), is the approximate number of days after the December solstice to January 1. This equation overestimates the declination near the September equinox by up to +1.5°. The sine function approximation by itself leads to an error of up to 0.26° and has been discouraged for use in solar energy applications. The 1971 Spencer formula (based on a
3132:
3084:
3120:
3096:
2254:
In addition to the annual north–south oscillation of the Sun's apparent position, corresponding to the variation of its declination described above, there is also a smaller but more complex oscillation in the east–west direction. This is caused by the tilt of the Earth's axis, and also by changes in
1684:
is small, its orbit can be approximated as a circle which causes up to 1° of error. The circle approximation means the EL would be 90° ahead of the solstices in Earth's orbit (at the equinoxes), so that sin(EL) can be written as sin(90+NDS)=cos(NDS) where NDS is the number of days after the
December
2227:
of light in the atmosphere, which causes the apparent angle of elevation of the Sun as seen by an observer to be higher than the actual angle of elevation, especially at low Sun elevations. For example, when the Sun is at an elevation of 10°, it appears to be at 10.1°. The Sun's declination can be
2209:
More complicated algorithms correct for changes to the ecliptic longitude by using terms in addition to the 1st-order eccentricity correction above. They also correct the 23.44° obliquity which changes very slightly with time. Corrections may also include the effects of the moon in offsetting the
1536:
by astronomers) is the angle between the Earth's axis and a line perpendicular to the Earth's orbit. The Earth's axial tilt changes slowly over thousands of years but its current value of about ε = 23°44' is nearly constant, so the change in solar declination during one year is nearly the same as
1489:
If the 90° axial tilt is decreased, then the absolute maximum and minimum values of the declination would decrease, to equal the axial tilt. Also, the shapes of the maxima and minima on the graph would become less acute, being curved to resemble the maxima and minima of a sine wave. However, even
2600:
Zhang, T., Stackhouse, P.W., Macpherson, B., and
Mikovitz, J.C., 2021. A solar azimuth formula that renders circumstantial treatment unnecessary without compromising mathematical rigor: Mathematical setup, application and extension of a formula based on the subsolar point and atan2 function.
2194:
of the Earth's orbit. The eccentricity varies very slowly over time, but for dates fairly close to the present, it can be considered to be constant. The largest errors in this equation are less than ± 0.2°, but are less than ± 0.03° for a given year if the number 10 is adjusted up or down in
2339:, plotted horizontally. Usually, the scales are chosen so that equal distances on the diagram represent equal angles in both directions on the celestial sphere. Thus 4 minutes (more precisely 3 minutes, 56 seconds), in the equation of time, are represented by the same distance as 1° in the
2195:
fractional days as determined by how far the previous year's
December solstice occurred before or after noon on December 22. These accuracies are compared to NOAA's advanced calculations which are based on the 1999 Jean Meeus algorithm that is accurate to within 0.01°.
2274:, using the word "equation" in a somewhat archaic sense meaning "correction". The oscillation is measured in units of time, minutes and seconds, corresponding to the amount that a sundial would be ahead of a clock. The equation of time can be positive or negative.
2214:
is applied, which depends on the observer's distance away from the center of the Earth. This correction is less than 0.0025°. The error in calculating the position of the center of the Sun can be less than 0.00015°. For comparison, the Sun's width is about 0.5°.
1477:
with time. Eventually, the Sun would be directly above the South Pole, with a declination of −90°; then it would start to move northward at a constant speed. Thus, the graph of solar declination, as seen from this highly tilted Earth, would resemble a
2255:
the speed of its orbital motion around the Sun produced by the elliptical shape of the orbit. The principal effects of this east–west oscillation are variations in the timing of events such as sunrise and sunset, and in the reading of a
2369:
Some analemmas are marked to show the position of the Sun on the graph on various dates, a few days apart, throughout the year. This enables the analemma to be used to make simple analog computations of quantities such as the times and
1378:
The path of the Sun over the celestial sphere through the course of the day for an observer at 56°N latitude. The Sun's path changes with its declination during the year. The intersections of the curves with the horizontal axis show
1071:
612:
1787:
787:
1134:
952:
722:
434:
375:
2177:
2016:
1671:
1278:
1233:
312:
1188:
2186:−1) and can include decimals to adjust for local times later or earlier in the day. The number 2, in (N-2), is the approximate number of days after January 1 to the Earth's
890:
996:
814:
648:
972:
1512:
The graph of apparent solar declination is therefore different in several ways from a sine wave. Calculating it accurately involves some complexity, as shown below.
834:
3051:
2974:
2210:
Earth's position from the center of the pair's orbit around the Sun. After obtaining the declination relative to the center of the Earth, a further correction for
1807:
The declination can be more accurately calculated by not making the two approximations, using the parameters of the Earth's orbit to more accurately estimate EL:
1348:
1324:
1300:
854:
525:
505:
477:
457:
2244:
The equation of time — above the axis a sundial will appear fast relative to a clock showing local mean time, and below the axis a sundial will appear slow.
1008:
2803:
2904:
3157:
2629:
2324:, which resembles a figure-8. An analemma can be pictured by superimposing photographs taken at the same time of day, a few days apart for a
544:
176:, for the observer's local time and location. This is the coordinate system normally used to calculate the position of the Sun in terms of
2861:
2833:
1691:
2693:
2517:
743:
2320:
is also occasionally, but rarely, used in other contexts.) It can be considered as an image of the Sun's apparent motion during a
2232:, to calculate its azimuth and also its true elevation, which can then be corrected for refraction to give its apparent position.
1086:
904:
266:, the number of days (positive or negative, including fractional days) since Greenwich noon, Terrestrial Time, on 1 January 2000 (
30:
3008:
1685:
solstice. By also using the approximation that arcsin is close to d·cos(NDS), the following frequently used formula is obtained:
2283:
384:
of the Sun (actually, of the Earth in its orbit around the Sun, but it is convenient to pretend the Sun orbits the Earth), is:
1544:, the angle between the rays of the Sun and the plane of the Earth's equator reaches its maximum value of 23°44'. Therefore, δ
2418:
2423:
668:
390:
331:
1490:
when the axial tilt equals that of the actual Earth, the maxima and minima remain more acute than those of a sine wave.
1364:
865:
173:
166:
90:
975:
154:
To find the Sun's position for a given location at a given time, one may therefore proceed in three steps as follows:
2027:
41:
1813:
1576:
1239:
1194:
3074:
3018:
861:
655:
618:
535:
159:
3022:
280:
3061:
2726:
2709:
1002:
To get RA at the right quadrant on computer programs use double argument Arctan function such as ATAN2(y,x)
2652:
247:, to a precision of about 0°.01 (36″), for dates between 1950 and 2050. Similar equations are coded into a
2224:
480:
2749:
2433:
2903:
Blanco-Muriel, Manuel; Alarcón-Padilla, Diego C; López-Moratalla, Teodoro; Lara-Coira, Martín (2001).
1158:
2919:
2776:
Sproul, Alistair B. (2007). "Derivation of the solar geometric relationships using vector analysis".
2564:
2482:
2191:
1681:
658:
of the Sun never exceeds 0.00033° (a little over 1″), and the distance of the Sun from the Earth, in
235:
144:
2807:
1434:
of 23.44°, but one lobe of the wave is several days longer than the other, among other differences.
3136:
2474:
2428:
2408:
2395:
2394:
We see light from the Sun about 20 angle seconds from where the Sun is when the light is seen. See
2344:
322:
256:
181:
124:
3124:
3112:
3057:
2741:
2580:
2554:
1677:
1470:
528:
252:
177:
875:
2689:
2625:
2542:
2513:
1560:
1412:
1392:
1149:
981:
799:
659:
627:
267:
244:
2857:
2683:
1528:, is the angle between the rays of the Sun and the plane of the Earth's equator. The Earth's
957:
3088:
2970:
2927:
2785:
2619:
2572:
2453:
2438:
2336:
2313:
2295:
2249:
2203:
2199:
1494:
819:
114:
98:
2762:
2656:
2264:
2229:
1388:
1145:
895:
2182:
N is the number of days since midnight UT as
January 1 begins (i.e. the days part of the
3038:
2923:
2568:
2618:
U.S. Naval
Observatory; U.K. Hydrographic Office, H.M. Nautical Almanac Office (2008).
1801:
1793:
1498:
1462:
1442:
1351:
1333:
1309:
1285:
839:
510:
490:
462:
442:
318:
128:
2931:
2350:
An analemma is drawn as it would be seen in the sky by an observer looking upward. If
17:
3146:
3067:
2584:
2576:
2321:
2198:(The above formula is related to a reasonably simple and accurate calculation of the
1479:
1446:
1408:
1396:
1327:
35:
2682:
U.S. Naval
Observatory Nautical Almanac Office (1992). P. Kenneth Seidelmann (ed.).
3152:
3100:
2478:
2316:, relative to its mean position, as seen from a fixed location on Earth. (The word
2183:
1797:
1445:
around the Sun, and if its axis were tilted 90°, so that the axis itself is on the
381:
213:
3064:
website. Positions of Solar System objects based on the INPOP series ephemerides.
2789:
2645:
2644:
Much the same set of equations, covering the years 1800 to 2200, can be found at
2882:
2340:
2291:
1486:
between plus and minus 90°, with linear segments between the maxima and minima.
1415:, when its value is the negative of the axial tilt. This variation produces the
1400:
1374:
1077:
271:
209:
205:
140:
110:
2606:
2240:
2223:
The declination calculations described above do not include the effects of the
2946:
2505:
2443:
2187:
1529:
1502:
1466:
1458:
1423:
1404:
869:
734:
197:
3028:
56:
43:
2829:
2660:
2382:. Analemmas without date markings are used to correct the time indicated by
1431:
1427:
1066:{\displaystyle \alpha =\arctan 2(\cos \epsilon \sin \lambda ,\cos \lambda )}
201:
193:
148:
2362:. This is usually done even when the analemma is marked on a geographical
2312:
is a diagram that shows the annual variation of the Sun's position on the
2270:
Since the main effect of this oscillation concerns time, it is called the
2448:
2413:
2309:
2303:
2287:
2211:
1541:
1506:
1438:
185:
136:
132:
118:
1411:, then decreases until reaching its minimum (−23.44° or -23°26') on the
2994:
2383:
2375:
2371:
2256:
1556:
1380:
1303:
248:
217:
3050:
website. Very accurate positions of Solar System objects based on the
2379:
1483:
1454:
1450:
1416:
607:{\displaystyle \lambda =L+1.915^{\circ }\sin g+0.020^{\circ }\sin 2g}
3047:
3043:
2282:
1680:(essentially, the Earth's position in its orbit). Since the Earth's
3095:
27:
Calculating the Sun's location in the sky at a given time and place
2659:. Graphs of the error of these equations, compared to an accurate
2559:
2366:, on which the continents, etc., are shown with west to the left.
2363:
2351:
2281:
2260:
2239:
1373:
94:
29:
1473:
at a constant rate, so that the solar declination would decrease
1461:, so its declination would be +90°. For the next few months, the
3032:
2649:
2355:
2325:
1782:{\displaystyle \delta _{\odot }=-23.44^{\circ }\cdot \cos \left}
106:
86:
3025:. An interactive calculator showing the Sun's path in the sky.
2332:
1792:
where N is the day of the year beginning with N=0 at midnight
1521:
857:
240:
102:
82:
3012:
782:{\displaystyle \epsilon =23.439^{\circ }-0.0000004^{\circ }n}
2347:
at a mean speed of 1° every 4 minutes, relative to the Sun.
1570:
The Sun's declination at any given moment is calculated by:
1437:
The following phenomena would occur if Earth were a perfect
1129:{\displaystyle \delta =\arcsin(\sin \epsilon \sin \lambda )}
947:{\displaystyle \alpha =\arctan(\cos \epsilon \tan \lambda )}
2947:"Solar Position Algorithm for Solar Radiation Applications"
239:, can be used to calculate the apparent coordinates of the
2742:"Fourier series representation of the position of the sun"
2688:. University Science Books, Mill Valley, CA. p. 12.
1387:
The Sun appears to move northward during the northern
717:{\displaystyle R=1.00014-0.01671\cos g-0.00014\cos 2g}
429:{\displaystyle g=357.528^{\circ }+0.9856003^{\circ }n}
370:{\displaystyle L=280.460^{\circ }+0.9856474^{\circ }n}
131:, so that the Sun appears to move across the sky in a
3072:
2508:(1991). "Chapter 12: Transformation of Coordinates".
2331:
An analemma can also be considered as a graph of the
2030:
1816:
1694:
1579:
1336:
1312:
1288:
1242:
1197:
1161:
1089:
1011:
984:
960:
907:
878:
842:
822:
802:
746:
671:
630:
547:
513:
493:
465:
445:
393:
334:
283:
2021:
which can be simplified by evaluating constants to:
1796:(UT) as January 1 begins (i.e. the days part of the
1559:, the center of the Sun appears to pass through the
3031:, at the NOAA Earth System Research Laboratories's
1426:of the Sun's declination during a year resembles a
1383:
in degrees from North where the Sun rises and sets.
737:is not obtained elsewhere, it can be approximated:
184:, and the two parameters can be used to depict the
2685:Explanatory Supplement to the Astronomical Almanac
2171:
2010:
1781:
1665:
1342:
1318:
1294:
1272:
1227:
1182:
1128:
1065:
990:
966:
946:
884:
848:
828:
808:
781:
716:
642:
606:
519:
499:
471:
451:
428:
369:
306:
1403:reaches a maximum equal to the angle of Earth's
479:in the range 0° to 360° by adding or subtracting
251:90 routine in Ref. and are used to calculate the
2190:. The number 0.0167 is the current value of the
2975:National Oceanic and Atmospheric Administration
2477:, the Earth's orbit would be stated as being a
2172:{\displaystyle \delta _{\odot }=-\arcsin \left}
1548:= +23°44' at the northern summer solstice and δ
1501:. Earth moves more rapidly around the Sun near
3058:General ephemerides of the solar system bodies
3039:NOAA's declination and sun position calculator
2011:{\displaystyle \delta _{\odot }=\arcsin \left}
1666:{\displaystyle \delta _{\odot }=\arcsin \left}
1273:{\displaystyle Z=R\sin \epsilon \sin \lambda }
1228:{\displaystyle Y=R\cos \epsilon \sin \lambda }
109:, the Sun appears to move with respect to the
2727:"Selected Astronomical Constants, 2015 (TXT)"
2710:"Selected Astronomical Constants, 2015 (PDF)"
1453:). At one date in the year, the Sun would be
8:
3011:, at National Renewable Energy Laboratory's
2729:. US Naval Observatory. 2014. p. K6–K7.
2715:. US Naval Observatory. 2014. p. K6–K7.
2607:https://doi.org/10.1016/j.renene.2021.03.047
2852:
2850:
2596:
2594:
1552:= −23°44' at the southern summer solstice.
259:as observed from the surface of the Earth.
2624:. U.S. Govt. Printing Office. p. C5.
2621:The Astronomical Almanac for the Year 2010
2335:, usually plotted vertically, against the
1370:Declination of the Sun as seen from Earth
135:that depends on the observer's geographic
2558:
2545:(2013). "The Sun's position in the sky".
2537:
2535:
2533:
2531:
2529:
2129:
2108:
2076:
2035:
2029:
1963:
1957:
1929:
1923:
1890:
1884:
1859:
1821:
1815:
1741:
1735:
1715:
1699:
1693:
1622:
1584:
1578:
1335:
1311:
1287:
1241:
1196:
1160:
1088:
1010:
983:
959:
906:
877:
841:
821:
801:
770:
757:
745:
670:
629:
586:
564:
546:
512:
492:
464:
444:
417:
404:
392:
358:
345:
333:
307:{\displaystyle n=\mathrm {JD} -2451545.0}
290:
282:
3079:
2497:
2466:
2971:"Atmospheric Refraction Approximation"
2758:
2747:
1148:rectangular equatorial coordinates in
7:
2512:. Richmond, VA: Willmann Bell, Inc.
274:for the desired time is known, then
158:calculate the Sun's position in the
3068:Solar position in R. Insol package.
2945:Ibrahim Reda & Afshin Andreas.
117:, along a circular path called the
2862:Earth System Research Laboratories
2834:Earth System Research Laboratories
1141:Rectangular equatorial coordinates
294:
291:
73:a.m. local time on a March morning
25:
245:mean equinox and ecliptic of date
3130:
3118:
3106:
3094:
3082:
1469:at constant speed, crossing the
1302:axis is in the direction of the
1183:{\displaystyle X=R\cos \lambda }
856:form a complete position of the
864:. This can be converted to the
3013:Renewable Resource Data Center
2650:U.S. Naval Observatory website
2419:Effect of Sun angle on climate
1505:, in early January, than near
1123:
1102:
1060:
1027:
941:
920:
321:of the Sun, corrected for the
192:This calculation is useful in
1:
2932:10.1016/s0038-092x(00)00156-0
2646:Approximate Solar Coordinates
3158:Dynamics of the Solar System
2905:"Computing the Solar Vector"
2790:10.1016/j.renene.2006.05.001
1365:Horizontal coordinate system
866:equatorial coordinate system
527:are really to be evaluated (
174:horizontal coordinate system
167:equatorial coordinate system
2858:"Solar Calculation Details"
2547:European Journal of Physics
2424:Newcomb's Tables of the Sun
3174:
3033:Global Monitoring Division
2577:10.1088/0143-0807/34/3/633
2354:is shown at the top, then
2301:
2247:
1407:(23.44° or 23°26') on the
1362:
862:ecliptic coordinate system
233:These equations, from the
160:ecliptic coordinate system
147:depends on the geographic
85:is a function of both the
2883:"Astronomical Algorithms"
1534:obliquity of the ecliptic
1482:rather than a sine wave,
885:{\displaystyle \epsilon }
870:obliquity of the ecliptic
735:obliquity of the ecliptic
729:Obliquity of the ecliptic
487:— which is to say,
3009:Solar Position Algorithm
991:{\displaystyle \lambda }
809:{\displaystyle \lambda }
643:{\displaystyle \beta =0}
139:. The time when the Sun
3019:Sun Position Calculator
2830:"NOAA Solar Calculator"
2605:, 172, 1333-1340. DOI:
2510:Astronomical Algorithms
2409:Solar annual aberration
2396:Solar annual aberration
1520:The declination of the
967:{\displaystyle \alpha }
2757:Cite journal requires
2740:J. W. Spencer (1971).
2299:
2245:
2219:Atmospheric refraction
2173:
2012:
1783:
1667:
1555:At the moment of each
1537:during the next year.
1465:would move toward the
1384:
1359:Horizontal coordinates
1344:
1320:
1296:
1274:
1229:
1184:
1130:
1067:
992:
968:
948:
886:
850:
830:
829:{\displaystyle \beta }
810:
793:Equatorial coordinates
783:
718:
644:
621:of the Sun is nearly:
608:
521:
501:
473:
453:
430:
371:
308:
127:about its axis causes
74:
18:Declination of the Sun
3029:NOAA Solar Calculator
2663:, can also be viewed.
2434:Solar elevation angle
2285:
2243:
2228:used, along with its
2202:, which is described
2174:
2013:
1784:
1668:
1377:
1363:Further information:
1345:
1321:
1297:
1275:
1230:
1185:
1131:
1068:
993:
969:
949:
887:
851:
831:
811:
784:
719:
645:
609:
522:
502:
474:
454:
431:
372:
309:
262:Start by calculating
105:over the course of a
33:
2672:Meeus (1991), p. 152
2483:orbital eccentricity
2028:
1814:
1692:
1682:orbital eccentricity
1577:
1352:North celestial pole
1334:
1310:
1286:
1240:
1195:
1159:
1087:
1009:
982:
958:
905:
876:
840:
820:
800:
744:
669:
628:
545:
511:
491:
463:
443:
391:
332:
281:
236:Astronomical Almanac
229:Ecliptic coordinates
224:Approximate position
2924:2001SoEn...70..431B
2569:2013EJPh...34..633J
2475:celestial mechanics
2429:Solar azimuth angle
1471:circles of latitude
892:, and continuing:
868:by calculating the
323:aberration of light
257:solar azimuth angle
182:solar azimuth angle
91:geographic location
79:position of the Sun
53: /
2995:Sundial#Noon marks
2655:2016-01-31 at the
2543:Jenkins, Alejandro
2390:Light-time effects
2300:
2298:to the same scale
2246:
2169:
2008:
1779:
1678:ecliptic longitude
1663:
1385:
1340:
1316:
1292:
1270:
1225:
1180:
1150:astronomical units
1126:
1063:
988:
964:
944:
882:
846:
826:
806:
779:
714:
660:astronomical units
640:
604:
536:ecliptic longitude
517:
497:
469:
449:
426:
367:
304:
253:solar zenith angle
178:solar zenith angle
93:of observation on
75:
2631:978-0-7077-4082-9
2333:Sun's declination
1972:
1938:
1899:
1750:
1561:celestial equator
1455:directly overhead
1413:December solstice
1393:celestial equator
1350:axis towards the
1343:{\displaystyle Z}
1319:{\displaystyle Y}
1295:{\displaystyle X}
849:{\displaystyle R}
656:ecliptic latitude
619:ecliptic latitude
520:{\displaystyle g}
500:{\displaystyle L}
483:of 360° as needed
472:{\displaystyle g}
452:{\displaystyle L}
16:(Redirected from
3165:
3135:
3134:
3133:
3123:
3122:
3121:
3111:
3110:
3109:
3099:
3098:
3087:
3086:
3085:
3078:
2997:
2992:
2986:
2985:
2983:
2981:
2967:
2961:
2960:
2958:
2956:
2951:
2942:
2936:
2935:
2909:
2900:
2894:
2893:
2891:
2889:
2879:
2873:
2872:
2870:
2868:
2854:
2845:
2844:
2842:
2840:
2826:
2820:
2819:
2817:
2815:
2806:. Archived from
2800:
2794:
2793:
2784:(7): 1187–1205.
2778:Renewable Energy
2773:
2767:
2766:
2760:
2755:
2753:
2745:
2737:
2731:
2730:
2723:
2717:
2716:
2714:
2706:
2700:
2699:
2679:
2673:
2670:
2664:
2642:
2636:
2635:
2615:
2609:
2603:Renewable Energy
2598:
2589:
2588:
2562:
2539:
2524:
2523:
2502:
2486:
2471:
2454:Sunrise equation
2439:Solar irradiance
2337:equation of time
2314:celestial sphere
2296:equation of time
2272:equation of time
2259:compared with a
2250:Equation of time
2236:Equation of time
2200:Equation of Time
2178:
2176:
2175:
2170:
2168:
2164:
2163:
2159:
2158:
2154:
2153:
2149:
2134:
2133:
2113:
2112:
2100:
2096:
2081:
2080:
2040:
2039:
2017:
2015:
2014:
2009:
2007:
2003:
2002:
1998:
1997:
1993:
1992:
1988:
1973:
1968:
1967:
1958:
1939:
1934:
1933:
1924:
1919:
1915:
1900:
1895:
1894:
1885:
1869:
1865:
1864:
1863:
1826:
1825:
1788:
1786:
1785:
1780:
1778:
1774:
1773:
1769:
1751:
1746:
1745:
1736:
1720:
1719:
1704:
1703:
1676:where EL is the
1672:
1670:
1669:
1664:
1662:
1658:
1657:
1653:
1632:
1628:
1627:
1626:
1589:
1588:
1349:
1347:
1346:
1341:
1325:
1323:
1322:
1317:
1301:
1299:
1298:
1293:
1279:
1277:
1276:
1271:
1234:
1232:
1231:
1226:
1189:
1187:
1186:
1181:
1135:
1133:
1132:
1127:
1072:
1070:
1069:
1064:
997:
995:
994:
989:
973:
971:
970:
965:
953:
951:
950:
945:
891:
889:
888:
883:
855:
853:
852:
847:
835:
833:
832:
827:
815:
813:
812:
807:
788:
786:
785:
780:
775:
774:
762:
761:
723:
721:
720:
715:
649:
647:
646:
641:
613:
611:
610:
605:
591:
590:
569:
568:
526:
524:
523:
518:
506:
504:
503:
498:
486:
478:
476:
475:
470:
458:
456:
455:
450:
435:
433:
432:
427:
422:
421:
409:
408:
376:
374:
373:
368:
363:
362:
350:
349:
313:
311:
310:
305:
297:
125:Earth's rotation
115:celestial sphere
72:
68:
67:
65:
64:
63:
58:
57:8.283°N 98.600°E
54:
51:
50:
49:
46:
21:
3173:
3172:
3168:
3167:
3166:
3164:
3163:
3162:
3143:
3142:
3141:
3131:
3129:
3119:
3117:
3107:
3105:
3093:
3083:
3081:
3073:
3044:HORIZONS System
3023:pveducation.org
3005:
3000:
2993:
2989:
2979:
2977:
2969:
2968:
2964:
2954:
2952:
2949:
2944:
2943:
2939:
2907:
2902:
2901:
2897:
2887:
2885:
2881:
2880:
2876:
2866:
2864:
2856:
2855:
2848:
2838:
2836:
2828:
2827:
2823:
2813:
2811:
2810:on 9 March 2012
2802:
2801:
2797:
2775:
2774:
2770:
2756:
2746:
2739:
2738:
2734:
2725:
2724:
2720:
2712:
2708:
2707:
2703:
2696:
2681:
2680:
2676:
2671:
2667:
2657:Wayback Machine
2643:
2639:
2632:
2617:
2616:
2612:
2599:
2592:
2541:
2540:
2527:
2520:
2504:
2503:
2499:
2495:
2490:
2489:
2485:of less than 1.
2472:
2468:
2463:
2458:
2404:
2392:
2306:
2280:
2265:local mean time
2252:
2238:
2230:right ascension
2221:
2139:
2135:
2125:
2124:
2120:
2104:
2086:
2082:
2072:
2071:
2067:
2057:
2053:
2031:
2026:
2025:
1978:
1974:
1959:
1956:
1952:
1925:
1905:
1901:
1886:
1883:
1879:
1855:
1851:
1847:
1840:
1836:
1817:
1812:
1811:
1759:
1755:
1737:
1734:
1730:
1711:
1695:
1690:
1689:
1646:
1642:
1618:
1614:
1610:
1603:
1599:
1580:
1575:
1574:
1566:
1551:
1547:
1527:
1518:
1391:, crossing the
1372:
1367:
1361:
1332:
1331:
1308:
1307:
1284:
1283:
1238:
1237:
1193:
1192:
1157:
1156:
1143:
1085:
1084:
1007:
1006:
980:
979:
974:is in the same
956:
955:
903:
902:
896:Right ascension
874:
873:
838:
837:
818:
817:
798:
797:
795:
766:
753:
742:
741:
731:
667:
666:
626:
625:
582:
560:
543:
542:
538:of the Sun is:
509:
508:
489:
488:
484:
461:
460:
441:
440:
413:
400:
389:
388:
354:
341:
330:
329:
279:
278:
231:
226:
172:convert to the
165:convert to the
143:the observer's
97:'s surface. As
70:
61:
59:
55:
52:
47:
44:
42:
40:
39:
28:
23:
22:
15:
12:
11:
5:
3171:
3169:
3161:
3160:
3155:
3145:
3144:
3140:
3139:
3127:
3115:
3103:
3091:
3071:
3070:
3065:
3055:
3041:
3036:
3026:
3016:
3004:
3003:External links
3001:
2999:
2998:
2987:
2962:
2937:
2918:(5): 431–441.
2895:
2874:
2846:
2821:
2795:
2768:
2759:|journal=
2732:
2718:
2701:
2694:
2674:
2665:
2637:
2630:
2610:
2590:
2553:(3): 633–652.
2525:
2518:
2496:
2494:
2491:
2488:
2487:
2465:
2464:
2462:
2459:
2457:
2456:
2451:
2446:
2441:
2436:
2431:
2426:
2421:
2416:
2411:
2405:
2403:
2400:
2391:
2388:
2302:Main article:
2279:
2276:
2248:Main article:
2237:
2234:
2220:
2217:
2180:
2179:
2167:
2162:
2157:
2152:
2148:
2145:
2142:
2138:
2132:
2128:
2123:
2119:
2116:
2111:
2107:
2103:
2099:
2095:
2092:
2089:
2085:
2079:
2075:
2070:
2066:
2063:
2060:
2056:
2052:
2049:
2046:
2043:
2038:
2034:
2019:
2018:
2006:
2001:
1996:
1991:
1987:
1984:
1981:
1977:
1971:
1966:
1962:
1955:
1951:
1948:
1945:
1942:
1937:
1932:
1928:
1922:
1918:
1914:
1911:
1908:
1904:
1898:
1893:
1889:
1882:
1878:
1875:
1872:
1868:
1862:
1858:
1854:
1850:
1846:
1843:
1839:
1835:
1832:
1829:
1824:
1820:
1802:Fourier series
1794:Universal Time
1790:
1789:
1777:
1772:
1768:
1765:
1762:
1758:
1754:
1749:
1744:
1740:
1733:
1729:
1726:
1723:
1718:
1714:
1710:
1707:
1702:
1698:
1674:
1673:
1661:
1656:
1652:
1649:
1645:
1641:
1638:
1635:
1631:
1625:
1621:
1617:
1613:
1609:
1606:
1602:
1598:
1595:
1592:
1587:
1583:
1564:
1549:
1545:
1525:
1517:
1514:
1463:subsolar point
1443:circular orbit
1371:
1368:
1360:
1357:
1356:
1355:
1339:
1315:
1291:
1280:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1235:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1190:
1179:
1176:
1173:
1170:
1167:
1164:
1142:
1139:
1138:
1137:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1074:
1073:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1000:
999:
987:
963:
943:
940:
937:
934:
931:
928:
925:
922:
919:
916:
913:
910:
881:
845:
825:
805:
794:
791:
790:
789:
778:
773:
769:
765:
760:
756:
752:
749:
730:
727:
726:
725:
713:
710:
707:
704:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
652:
651:
639:
636:
633:
615:
614:
603:
600:
597:
594:
589:
585:
581:
578:
575:
572:
567:
563:
559:
556:
553:
550:
516:
496:
468:
448:
437:
436:
425:
420:
416:
412:
407:
403:
399:
396:
378:
377:
366:
361:
357:
353:
348:
344:
340:
337:
319:mean longitude
315:
314:
303:
300:
296:
293:
289:
286:
230:
227:
225:
222:
190:
189:
170:
163:
129:diurnal motion
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3170:
3159:
3156:
3154:
3151:
3150:
3148:
3138:
3128:
3126:
3116:
3114:
3104:
3102:
3097:
3092:
3090:
3080:
3076:
3069:
3066:
3063:
3059:
3056:
3053:
3052:JPL DE series
3049:
3045:
3042:
3040:
3037:
3034:
3030:
3027:
3024:
3020:
3017:
3014:
3010:
3007:
3006:
3002:
2996:
2991:
2988:
2976:
2972:
2966:
2963:
2948:
2941:
2938:
2933:
2929:
2925:
2921:
2917:
2913:
2906:
2899:
2896:
2884:
2878:
2875:
2863:
2859:
2853:
2851:
2847:
2835:
2831:
2825:
2822:
2809:
2805:
2799:
2796:
2791:
2787:
2783:
2779:
2772:
2769:
2764:
2751:
2743:
2736:
2733:
2728:
2722:
2719:
2711:
2705:
2702:
2697:
2695:0-935702-68-7
2691:
2687:
2686:
2678:
2675:
2669:
2666:
2662:
2658:
2654:
2651:
2647:
2641:
2638:
2633:
2627:
2623:
2622:
2614:
2611:
2608:
2604:
2597:
2595:
2591:
2586:
2582:
2578:
2574:
2570:
2566:
2561:
2556:
2552:
2548:
2544:
2538:
2536:
2534:
2532:
2530:
2526:
2521:
2519:0-943396-35-2
2515:
2511:
2507:
2501:
2498:
2492:
2484:
2480:
2476:
2470:
2467:
2460:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2427:
2425:
2422:
2420:
2417:
2415:
2412:
2410:
2407:
2406:
2401:
2399:
2397:
2389:
2387:
2385:
2381:
2377:
2373:
2367:
2365:
2361:
2357:
2353:
2348:
2346:
2345:Earth rotates
2342:
2338:
2334:
2329:
2327:
2323:
2319:
2315:
2311:
2305:
2297:
2293:
2289:
2284:
2277:
2275:
2273:
2268:
2266:
2262:
2258:
2251:
2242:
2235:
2233:
2231:
2226:
2218:
2216:
2213:
2207:
2205:
2201:
2196:
2193:
2189:
2185:
2165:
2160:
2155:
2150:
2146:
2143:
2140:
2136:
2130:
2126:
2121:
2117:
2114:
2109:
2105:
2101:
2097:
2093:
2090:
2087:
2083:
2077:
2073:
2068:
2064:
2061:
2058:
2054:
2050:
2047:
2044:
2041:
2036:
2032:
2024:
2023:
2022:
2004:
1999:
1994:
1989:
1985:
1982:
1979:
1975:
1969:
1964:
1960:
1953:
1949:
1946:
1943:
1940:
1935:
1930:
1926:
1920:
1916:
1912:
1909:
1906:
1902:
1896:
1891:
1887:
1880:
1876:
1873:
1870:
1866:
1860:
1856:
1852:
1848:
1844:
1841:
1837:
1833:
1830:
1827:
1822:
1818:
1810:
1809:
1808:
1805:
1803:
1799:
1795:
1775:
1770:
1766:
1763:
1760:
1756:
1752:
1747:
1742:
1738:
1731:
1727:
1724:
1721:
1716:
1712:
1708:
1705:
1700:
1696:
1688:
1687:
1686:
1683:
1679:
1659:
1654:
1650:
1647:
1643:
1639:
1636:
1633:
1629:
1623:
1619:
1615:
1611:
1607:
1604:
1600:
1596:
1593:
1590:
1585:
1581:
1573:
1572:
1571:
1568:
1562:
1558:
1553:
1543:
1538:
1535:
1531:
1523:
1515:
1513:
1510:
1508:
1504:
1500:
1496:
1495:Earth's orbit
1491:
1487:
1485:
1481:
1480:triangle wave
1476:
1472:
1468:
1464:
1460:
1456:
1452:
1448:
1447:orbital plane
1444:
1440:
1435:
1433:
1429:
1425:
1420:
1418:
1414:
1410:
1409:June solstice
1406:
1402:
1398:
1397:March equinox
1394:
1390:
1382:
1376:
1369:
1366:
1358:
1353:
1337:
1329:
1328:June Solstice
1326:axis towards
1313:
1305:
1304:March equinox
1289:
1281:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1236:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1191:
1177:
1174:
1171:
1168:
1165:
1162:
1155:
1154:
1153:
1151:
1147:
1140:
1120:
1117:
1114:
1111:
1108:
1105:
1099:
1096:
1093:
1090:
1083:
1082:
1081:
1079:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1024:
1021:
1018:
1015:
1012:
1005:
1004:
1003:
985:
977:
961:
938:
935:
932:
929:
926:
923:
917:
914:
911:
908:
901:
900:
899:
897:
893:
879:
871:
867:
863:
859:
843:
823:
803:
792:
776:
771:
767:
763:
758:
754:
750:
747:
740:
739:
738:
736:
728:
711:
708:
705:
702:
699:
696:
693:
690:
687:
684:
681:
678:
675:
672:
665:
664:
663:
661:
657:
637:
634:
631:
624:
623:
622:
620:
601:
598:
595:
592:
587:
583:
579:
576:
573:
570:
565:
561:
557:
554:
551:
548:
541:
540:
539:
537:
534:Finally, the
532:
530:
514:
494:
482:
466:
446:
423:
418:
414:
410:
405:
401:
397:
394:
387:
386:
385:
383:
364:
359:
355:
351:
346:
342:
338:
335:
328:
327:
326:
324:
320:
301:
298:
287:
284:
277:
276:
275:
273:
269:
265:
260:
258:
254:
250:
246:
242:
238:
237:
228:
223:
221:
219:
215:
211:
207:
203:
199:
195:
187:
183:
179:
175:
171:
168:
164:
161:
157:
156:
155:
152:
150:
146:
142:
138:
134:
130:
126:
122:
120:
116:
112:
108:
104:
100:
96:
92:
88:
84:
80:
66:
62:8.283; 98.600
38:in Thailand (
37:
36:Phang Nga Bay
34:The Sun over
32:
19:
3137:Solar System
3054:ephemerides.
2990:
2978:. Retrieved
2965:
2953:. Retrieved
2940:
2915:
2912:Solar Energy
2911:
2898:
2886:. Retrieved
2877:
2865:. Retrieved
2837:. Retrieved
2824:
2812:. Retrieved
2808:the original
2798:
2781:
2777:
2771:
2750:cite journal
2735:
2721:
2704:
2684:
2677:
2668:
2640:
2620:
2613:
2602:
2550:
2546:
2509:
2500:
2479:Kepler orbit
2469:
2393:
2368:
2359:
2349:
2330:
2317:
2307:
2271:
2269:
2253:
2222:
2208:
2197:
2192:eccentricity
2184:ordinal date
2181:
2020:
1806:
1798:ordinal date
1791:
1675:
1569:
1554:
1539:
1533:
1532:(called the
1519:
1516:Calculations
1511:
1493:In reality,
1492:
1488:
1474:
1449:(similar to
1436:
1421:
1386:
1146:Right-handed
1144:
1075:
1001:
894:
796:
732:
653:
616:
533:
438:
382:mean anomaly
379:
316:
263:
261:
234:
232:
214:solar energy
191:
153:
123:
99:Earth orbits
78:
76:
3125:Outer space
3113:Spaceflight
2980:28 February
2955:28 February
2888:28 February
2867:28 February
2839:28 February
2814:28 February
2506:Meeus, Jean
2341:declination
2292:declination
2290:with solar
1401:declination
1078:declination
272:Julian date
210:climatology
206:meteorology
111:fixed stars
60: /
3147:Categories
2804:"SunAlign"
2493:References
2444:Solar time
2358:is to the
2225:refraction
2188:perihelion
1530:axial tilt
1503:perihelion
1499:elliptical
1484:zigzagging
1467:South Pole
1459:North Pole
1424:line graph
1405:axial tilt
1330:, and the
733:Where the
270:). If the
198:navigation
69:), at 7:00
3089:Astronomy
3060:, at the
3046:, at the
2661:ephemeris
2648:, at the
2585:119282288
2560:1208.1043
2144:−
2131:∘
2118:
2110:∘
2078:∘
2065:
2051:
2045:−
2037:⊙
2033:δ
1983:−
1965:∘
1950:
1941:⋅
1936:π
1931:∘
1892:∘
1877:
1871:⋅
1861:∘
1853:−
1845:
1834:
1823:⊙
1819:δ
1753:⋅
1743:∘
1728:
1722:⋅
1717:∘
1709:−
1701:⊙
1697:δ
1640:
1634:⋅
1624:∘
1616:−
1608:
1597:
1586:⊙
1582:δ
1542:solstices
1432:amplitude
1428:sine wave
1268:λ
1265:
1259:ϵ
1256:
1223:λ
1220:
1214:ϵ
1211:
1178:λ
1175:
1121:λ
1118:
1112:ϵ
1109:
1100:
1091:δ
1058:λ
1055:
1046:λ
1043:
1037:ϵ
1034:
1022:
1013:α
986:λ
962:α
939:λ
936:
930:ϵ
927:
918:
909:α
880:ϵ
824:β
804:λ
772:∘
768:0.0000004
764:−
759:∘
748:ϵ
706:
697:−
691:
682:−
632:β
596:
588:∘
574:
566:∘
549:λ
481:multiples
419:∘
415:0.9856003
406:∘
360:∘
356:0.9856474
347:∘
302:2451545.0
299:−
202:surveying
194:astronomy
149:longitude
3035:website.
3015:website.
2653:Archived
2481:with an
2449:Sun path
2414:Ecliptic
2402:See also
2384:sundials
2372:azimuths
2343:, since
2318:analemma
2310:analemma
2304:Analemma
2288:analemma
2278:Analemma
2263:showing
2212:parallax
1507:aphelion
1475:linearly
1430:with an
1381:azimuths
976:quadrant
954:, where
220:design.
186:Sun path
145:meridian
141:transits
137:latitude
133:Sun path
119:ecliptic
89:and the
3075:Portals
2920:Bibcode
2565:Bibcode
2376:sunrise
2257:sundial
2127:0.98565
2074:0.98565
2059:0.39779
1567:is 0°.
1563:, and δ
1557:equinox
1540:At the
1457:at the
1441:, in a
1417:seasons
1395:on the
860:in the
700:0.00014
685:0.01671
679:1.00014
654:as the
485:
402:357.528
343:280.460
268:J2000.0
249:Fortran
218:sundial
113:on the
81:in the
48:98°36′E
2692:
2628:
2583:
2516:
2380:sunset
2048:arcsin
1970:365.24
1944:0.0167
1897:365.24
1831:arcsin
1594:arcsin
1451:Uranus
1439:sphere
1399:. Its
1389:spring
1306:, the
1282:Where
1097:arcsin
1019:arctan
915:arctan
755:23.439
662:, is:
531:360).
325:, is:
216:, and
71:
45:8°17′N
3101:Stars
3062:IMCCE
3021:, at
2950:(PDF)
2908:(PDF)
2713:(PDF)
2581:S2CID
2555:arXiv
2461:Notes
2364:globe
2360:right
2352:north
2261:clock
2106:1.914
1857:23.44
1713:23.44
1620:23.44
1152:are:
584:0.020
562:1.915
169:, and
95:Earth
2982:2012
2957:2012
2890:2012
2869:2012
2841:2012
2816:2012
2763:help
2690:ISBN
2626:ISBN
2514:ISBN
2378:and
2356:west
2326:year
2322:year
2294:and
2204:here
1076:and
836:and
617:The
507:and
459:and
439:Put
380:The
317:The
255:and
180:and
107:year
101:the
87:time
77:The
3153:Sun
3048:JPL
2928:doi
2786:doi
2573:doi
2473:In
2374:of
2308:An
2286:An
2206:.)
2115:sin
2062:cos
1961:360
1947:sin
1927:360
1888:360
1874:cos
1842:sin
1748:365
1739:360
1725:cos
1637:sin
1605:sin
1524:, δ
1522:Sun
1497:is
1262:sin
1253:sin
1217:sin
1208:cos
1172:cos
1115:sin
1106:sin
1052:cos
1040:sin
1031:cos
978:as
933:tan
924:cos
858:Sun
703:cos
688:cos
593:sin
571:sin
529:mod
241:Sun
103:Sun
83:sky
3149::
2973:.
2926:.
2916:70
2914:.
2910:.
2860:.
2849:^
2832:.
2782:32
2780:.
2754::
2752:}}
2748:{{
2593:^
2579:.
2571:.
2563:.
2551:34
2549:.
2528:^
2398:.
2386:.
2328:.
2094:10
1913:10
1767:10
1422:A
1419:.
1080:,
898:,
872:,
816:,
243:,
212:,
208:,
204:,
200:,
196:,
151:.
121:.
3077::
2984:.
2959:.
2934:.
2930::
2922::
2892:.
2871:.
2843:.
2818:.
2792:.
2788::
2765:)
2761:(
2744:.
2698:.
2634:.
2587:.
2575::
2567::
2557::
2522:.
2166:]
2161:)
2156:)
2151:)
2147:2
2141:N
2137:(
2122:(
2102:+
2098:)
2091:+
2088:N
2084:(
2069:(
2055:[
2042:=
2005:]
2000:)
1995:)
1990:)
1986:2
1980:N
1976:(
1954:(
1921:+
1917:)
1910:+
1907:N
1903:(
1881:(
1867:)
1849:(
1838:[
1828:=
1776:]
1771:)
1764:+
1761:N
1757:(
1732:[
1706:=
1660:]
1655:)
1651:L
1648:E
1644:(
1630:)
1612:(
1601:[
1591:=
1565:☉
1550:☉
1546:☉
1526:☉
1354:.
1338:Z
1314:Y
1290:X
1250:R
1247:=
1244:Z
1205:R
1202:=
1199:Y
1169:R
1166:=
1163:X
1136:.
1124:)
1103:(
1094:=
1061:)
1049:,
1028:(
1025:2
1016:=
998:,
942:)
921:(
912:=
844:R
777:n
751:=
724:.
712:g
709:2
694:g
676:=
673:R
650:,
638:0
635:=
602:g
599:2
580:+
577:g
558:+
555:L
552:=
515:g
495:L
467:g
447:L
424:n
411:+
398:=
395:g
365:n
352:+
339:=
336:L
295:D
292:J
288:=
285:n
264:n
188:.
162:,
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
